<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%Produce the formulae in the first columns of the second horizontal
%part of page 10
%
%The macrohas one parameter:
%       1) The size available to typeset the math
\newcommand\TTenSerieFour[1]{%
   \parbox[t]{#1}{%
      \TTenSeriesFontSize
      %This command typeset a line equation about the series on page 10
      %when the serie is in one column
      %
      %The command has 2 parameters
      %      1) The first formula to be expanded
      %      2) its expanded version
      \def\LineOfArray##1##2{%
        ##1&amp;=&amp;##2\\[\TTenExpansionSkip]%
      }
      %To have some longer vertical space between some lines
      %of the array (to better separate some equation).
      \deflength\VSpace{1.2\TTenExpansionSkip}
      $\displaystyle
       \begin{array}{l@{\hspace{.1em}}c@{\hspace{.2em}}l}
         \LineOfArray{\zeta(x)}
                     {\prod_p \frac{1}{1 - p^{-x}}}
         \LineOfArray{\zeta^2(x)}
                     {\sum_{i=1}^\infty \frac{d(i)}{x^i}} 
         %The previous line is too long
          &amp; &amp; \text{where }d(n) = \sum_{d \vert n} 1 \\[\VSpace]
         \LineOfArray{\zeta(x)\zeta(x-1)}
                     {\sum_{i=1}^\infty \frac{S(i)}{x^i}}
         %The previous line is too long
          &amp; &amp; \text{where }S(n) = \sum_{d \vert n} d \\[\VSpace]
         \LineOfArray{\zeta(2n)}
                     {\frac{2^{2n-1} \vert B_{2n} \vert}{(2n)!} \pi^{2n}, \quad n \in \Natural}
         \LineOfArray{\frac{x}{\sin x}}
                     {\sum_{i=0}^\infty (-1)^{i-1}\frac{(4^i - 2) B_{2i} x^{2i}}{(2i)!}}
         \LineOfArray{\left(\frac{1 - \sqrt{1-4x}}{2x}\right)^n}
                     {\sum_{i=0}^\infty \frac{n(2i + n - 1)!}{i! (n + i)!} x^i}
         \LineOfArray{e^x \sin x}%
                     {\sum_{i=1}^\infty \frac{2^{i/2} \sin \tfrac{i \pi}{4}}{i!} x^i}
         \LineOfArray{\frac{\sqrt{1 - \sqrt{1 -x}}}{x}}%
                     {\sum_{i=0}^\infty \frac{(4i)!}{16^i \sqrt{2} (2i)! (2i+1)!} x^i}
         \LineOfArray{\left(\frac{\arcsin x}{x}\right)^2}%
                     {\sum_{i=0}^\infty \frac{ 4^i i!^2}{(i + 1)(2i +1)!} x^{2i}}
      \end{array}$
   }
}
%The title of this part
%\newcommand\TTenSeriesBTitle{Series (continued)}
\newcommand\TTenSeriesBTitle{\relax}
</pre></body></html>